In the previous section we showed that our setup captures the shock dynamics and transient decay exhibited by the VBE. In that case, all solutions decay to zero at long times.
Now we examine the KSE, where the dynamics are chaotic and collapse onto a low-dimensional inertial manifold at long-times [1]}, [2]}, [3]}, [4]}. A useful model for the KSE should keep long-time trajectories on the attractor and provide a means for reduced-order modeling to capitalize on the low-dimensionality of the manifold on which the data lies.
